A053831 Sum of digits of n written in base 11.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 7, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 0

Author

Henry Bottomley, Mar 28 2000

Keywords

• base
• nonn
• easy

Comments

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8,9,10}, 1->{1,2,3,4,5,6,7,8,9,10,11}, 2->{2,3,4,5,6,7,8,9,10,11,12}, etc. - Robert G. Wilson v, Jul 27 2006

Examples

			a(20) = 1 + 9 = 10 because 20 is written as 19 base 11.
		

Links

• Tanar Ulric, Table of n, a(n) for n = 0..10000
• Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
• Robert Walker, Self Similar Sloth Canon Number Sequences.
• Eric Weisstein's World of Mathematics, Digit Sum.

Crossrefs

Cf. A007953, A064458, A138530.

Sum of digits of n written in bases 2-16: A000120, A053735, A053737, A053824, A053827, A053828, A053829, A053830, A007953, this sequence, A053832, A053833, A053834, A053835, A053836.

Programs

C

int Base11DigitSum(int n) {
   int count = 0;
   while (n != 0) { count += n % 11; n = n / 11; }
   return count;
} // Tanar Ulric, Oct 20 2021

Mathematica

Table[Plus @@ IntegerDigits[n, 11], {n, 0, 86}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 10}]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)

PARI

a(n)=if(n<1,0,if(n%11,a(n-1)+1,a(n/11)))

PARI

a(n)=sumdigits(n,11) \\ Charles R Greathouse IV, Oct 20 2021

Formula

From Benoit Cloitre, Dec 19 2002: (Start)

a(0)=0, a(11n+i) = a(n)+i for 0 <= i <= 10.

a(n) = n-(m-1)*(Sum_{k>0} floor(n/m^k)) = n-(m-1)*A064458(n). (End)

a(n) = A138530(n,11) for n > 10. - Reinhard Zumkeller, Mar 26 2008

Sum_{n>=1} a(n)/(n*(n+1)) = 11*log(11)/10 (Shallit, 1984). - Amiram Eldar, Jun 03 2021